Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007
Generating functions from the point of view of Rota–Baxter algebras Li Guo Abstract. We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a special free complete Rota-Baxter algebra. This allows us to use free Rota–Baxter algebras to give a wide class of algebraic structures where generalizations of generating functions can be studied. We illustrate this by several cases and examples.
1. Introduction The power series (or ordinary) generating function of a number sequence a0 , a1 , a2 , · · · is the power series X A(z) = an z n . n≥0
The exponential generating function (EGF) of a number sequence a1 , a2 , · · · is the exponential power series X zn B(z) = an . n! n≥1
For some sequences, the exponential series has a better formula. For example, the Bell numbers b(n) have exponential generating function X xn B(n) = exp(ex − 1). n! n≥0
The Bernoulli numbers Bn have exponential generating function X xn x Bn = x . n! e −1 n≥0
Such nice analytic expressions are not available in terms of power series generating functions. One way to view the power series generating functions and exponential generating functions in the same framework is that they each give a way to encode a number sequence an , n ≥ 0 as the coefficients of a linear combination with respect to a basis of the power series algebra R[[z]]. The basis is z n , n ≥ 0 for power series generating functions and is the divided powers z n /n!, n ≥ 0 for exponential generating functions. The distinction is the two different bases for the same power series ring. We can also view the difference externally: vk = z k , k ≥ 0 is the standard basis of the algebra Y A := R vk , with componentwise product vm vn = vm+n k≥0
2000 Mathematics Subject Classification. Primary 05A15; secondary 05A18, 16W99. Key words and phrases. exponential generating functions, shuffle products, free Rota-Baxter algebras, divided powers, Stirling numbers. The author is supported in part by NSF grant DMS-0505643.
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Li Guo
while wk = z k /k! is the standard basis of the divided power algebra Y m+n B := R wk , with componentwise product wm wn = wm+n . m
k≥0
We can take this point of view further and consider the following general framework for generating functions: A complete filtered R-algebra is a R-algebra A with ideals An , n ≥ 0, such that Am An ⊆ Am+n and A is complete with respect to the metric on A induced by An . In other words, the natural map A → lim A/An ←−
is bijective. Let U := {uj , j ∈ J}, be a basis of A that is compatible with its filtration in the sense that U ∩ Ak is a basis P of Ak , k ≥ 0. Then a U-generating function of a family of numbers aj ∈ R, j ∈ J, is the element j∈J aj uj in A. In this context, a power series generating function is a U-generating function when U is taken to be the basis vk = {z k , k ≥ 0} in the complete filtered algebra Y Y R[[z]] = R zk ∼ R vk = A = k≥0
k≥0
and an exponential generating function is a U-generating function when U is taken to be the basis {wk = z k /k!, k ≥ 0} in the complete filtered algebra Y zk Y ∼ R[[z]] = R wk = B. R = k! k≥0
k≥0
In both cases, the complete filtration on R[[z]] is given by the ideals z n R[[z]] which is also uk A (resp. vk B). Of course such a formal definition in such generality is of little use unless (1) it can be naturally related to the ordinary generating functions or exponential generating functions; (2) it is useful in the study of number sequences and number families. We will show that free Rota-Baxter algebras do give a generalization with these conditions. In Section 2 we review the construction of free commutative Rota–Baxter algebras and show that their completions give a large class of complete filtered algebras. In Section 3 we show that such complete free Rota-Baxter algebras give the exponential generating functions in a very special case. We then show how other instances of complete free Rota-Baxter algebras give rise to interesting generating functions of sequences of numbers or multi-indexed families of numbers, such as the Stirling numbers of the second kind and partition numbers. 2. Complete free commutative Rota-Baxter algebras Let λ ∈ R be a constant. A Rota–Baxter algebra of weight λ is a pair (R, P ) where R is a unitary k-algebra and P : R → R is a linear operator such that (2.1)
P (x)P (y) = P (xP (y)) + P (P (x)y) + λP (xy),
for any x, y ∈ R. Often θ = −λ is used, especially in the physics literature. Let (R, P ) and (R′ , P ′ ) be Rota-Baxter algebras of weight λ. A Rota-Baxter algebra homomorphism f from (R, P ) to (R′ , P ′ ) is an algebra homomorphism f : R → R′ such that f ◦ P = P ′ ◦ f . The study of Rota-Baxter algebras was started by G. Baxter in 1960 and was popularized largely by Rota and his school in 1960s and 70s and again in 1990s. In recently years, there have been several interesting developments of Rota–Baxter algebras in theoretical physics and mathematics, including quantum field theory, Yang–Baxter equations, shuffle products, operads, Hopf algebras, combinatorics and number theory. The most prominent of these is the work of Connes and Kreimer in their Hopf algebraic approach to renormalization theory in perturbative quantum field theory [8, 9, 13]. 2.1. Free commutative Rota-Baxter algebras. We recall the construction of free commutative Rota-Baxter algebras in terms of mixable shuffles. See [20, 21, 19] for details. Given a commutative algebra C which will often be taken as R, λ ∈ C, and a C-algebra A, the free commutative Rota-Baxter C-algebra on A is defined to be a Rota-Baxter C-algebra ( XC,λ (A), PA ) together with a C-algebra homomorphism jA : A → XC,λ (A) with the property that, for any Rota-Baxter Calgebra (R, P ) and any C-algebra homomorphism f : A → R, there is a unique Rota-Baxter C-algebra homomorphism f˜ : ( XC,λ (A), PA ) → (R, P ) such that jA ◦ f˜ = f as C-algebra homomorphisms.
GENERATING FUNCTIONS AND ROTA–BAXTER ALGEBRAS
3
One realization of this free commutative Rota-Baxter algebra is given by the mixable shuffle Rota-Baxter algebra. The mixable shuffle Rota-Baxter algebra is a pair ( XC,λ (A), PA ), where XC,λ (A) is a C-algebra in which • the C-module structure is given by the direct sum ∞ M A⊗n , where A⊗n = A ⊗C . . . ⊗C A; | {z } n=1
n−factors
• the multiplication is given by the augmented mixable shuffle product ⋄, recursively defined on A⊗m ⊗ A⊗n by a0 ⋄ (b0 ⊗ b1 ⊗ . . . ⊗ bn ) = (a0 ⊗ a1 ⊗ . . . ⊗ am ) ⋄ b0
=
a0 b 0 ⊗ b 1 ⊗ . . . ⊗ b n , a0 b0 ⊗ a1 ⊗ . . . ⊗ am , ai , bj ∈ A⊗1 = A,
and (a0 ⊗ a1 ⊗ . . . ⊗ am ) ⋄ (b0 ⊗ b1 ⊗ . . . ⊗ bn ) = (a0 b0 ) ⊗ (a1 ⊗ . . . ⊗ am ) ⋄ (1 ⊗ b1 ⊗ . . . ⊗ bn )
(2.2)
with the convention that
+(a0 b0 ) ⊗ (1 ⊗ a1 ⊗ . . . ⊗ am ) ⋄ (b1 ⊗ . . . ⊗ bn ) +λa0 b0 ⊗ (a1 ⊗ . . . ⊗ am ) ⋄ (b1 ⊗ . . . ⊗ bn ) , ai , bj ∈ A
a ⋄ (1 ⊗ b) = a ⊗ b, (1 ⊗ a) ⋄ b = b ⊗ a, a ⋄ b = ab, for a, b ∈ A. The Baxter operator PA is defined by PA (a1 ⊗ . . . ⊗ am ) = 1 ⊗ a1 ⊗ . . . ⊗ am , a1 ⊗ . . . ⊗ am ∈ A⊗m , m ≥ 1. Since the mixable shuffle product is compatible with the product on A, we will suppress the notation ⋄. We will also express C and λ from XC,λ (A) when there is not danger of confusion. Note that assuming PA is a Rota-Baxter operator and thus satisfies Eq. (2.1), then Eq. (2.2) follows. For example, we have (a0 ⊗ a1 ⊗ a2 )(b0 ⊗ b1 ) =
a0 b0 ⊗ ((a1 ⊗ a2 )(1 ⊗ b1 ) + b1 (1 ⊗ a1 ⊗ a2 ) + λ(a1 ⊗ a2 )b1 )
= =
a0 b0 ⊗ (a1 ⊗ (a2 (1 ⊗ b1 ) + (b1 (1 ⊗ a2 )) + λa2 b1 ) + b1 ⊗ a1 ⊗ a2 + a1 b1 ⊗ a2 ) a0 b0 ⊗ (a1 ⊗ a2 ⊗ b1 + a1 ⊗ b1 ⊗ a2 + λa1 ⊗ a2 b1 + b1 ⊗ a1 ⊗ a2 + λa1 b1 ⊗ a2 ) .
Theorem 2.1. ([20, Theorem 4.1]) For any C-algebra A, ( X(A), PA ), together with the natural embedding jA : A → X(A), is a free Baxter C-algebra on A (of weight λ) in the sense that the triple ( X(A), PA , jA ) satisfies the following universal property: For any Baxter C-algebra (R, P ) and any Calgebra map ϕ : A → R, there exists a unique Baxter C-algebra homomorphism ϕ˜ : ( X(A), PA ) → (R, P ) such that the diagram jA
/ X(A) A OOO OOO OOO ϕ ˜ ϕ OOOO OO' R commutes. Alternatively, can defined to be the tensor product algebra A ⊗ X+ (A) where the multiplication L X(A) + ⊗k on X (A) = k≥1 A is given in the explicit form by the mixable shuffle product [20] and in the recursive form by a generalization [10, 23] of the quasi-shuffle algebra defined by Hoffman [26] in the study of multiple zeta values. Quasi-shuffle is also known as harmonic product [25] and coincides with the stuffle product [2, 4] in the study of multiple zeta values. Variations of the stuffle product have also appeared in [6, 14]. It is shown [10] to be the same as the mixable shuffle product [20, 21] which is also call overlapping shuffles [24] and generalized shuffles [16], and can be interpreted in terms of Delannoy paths [1, 15, 28].
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Li Guo
2.2. Complete free Rota-Baxter algebras. For a given commutative algebra A, it is easy to see that the submodules M Filk X(A) := A⊗i , i ≥ 0 i≥k
of X(A) are ideals of X(A). They are in fact Rota-Baxter ideals of X(A) in the sense that PA (Filk X(A)) ⊆ Filk X(A). Further, ∩k≥0 Filk X(A) = 0. Thus Y d X(A) := lim X(A)/Filk X(A) ∼ A⊗k = ←−
k≥0
is a complete filtered algebra and contains X(A) as a subalgebra. It coincides with the complete free commutative Rota-Baxter algebra defined in [21]. 3. Generating functions from Rota-Baxter algebras
We first interpret exponential generating functions in terms of free commutative Rota-Baxter algebras. We then consider other cases where free commutative Rota-Baxter algebras give rise to generating functions. 3.1. Connection with exponential power series. Let A = R. Then Xλ (A) = Xλ (R) = ⊕k≥1 R⊗k = ⊕k≥1 R 1k ,
(3.1)
where 1k = 1 ⊗ · · · ⊗ 1. The augmented shuffle product in this special case is | {z } (k+1)−terms
(3.2)
1m ⋄ 1n =
m X
λk
i=0
When λ = 0, we have 1m ⋄ 1n =
m+n−k m
m+n m
m k
1m+n−k .
1m+n .
Q We then have d Xλ (R) = k≥0 R1k . This is also the cofree differential algebra [27] with the differential operator d(xn ) = xn−1 , d(1) = 0. Denote xn = xn /n! (divided powers). Then as an algebra,
with multiplication given by xm xn =
R[x] = ⊕n≥0 Rxn m+n xm+n . This extends to an isomorphism m
R[[x]] → d X(R) =
Y
R 1k ,
xk 7→ 1k , k ≥ 0.
k≥0
Through this isomorphism the theory of exponential generating function is translated to a theory of generating functions in d X(R), and can be generalized to d Xλ (A) for other algebras A and other weight λ. We will next demonstrate how this can be done in several cases. More systematic results will be provided in the full paper. 3.2. λ-exponential generating functions. We first consider free Rota-Baxter algebra X(R) on R of weight 1. It is given by Eq. (3.1) with product given by Eq. (3.2) with λ = 1. We quote from [32], Proposition 5.1.1, the following simple yet fundamental property of exponential generating functions which underlies the prominent P role played by these generating functions. See [32] for details. For any function f : N → R, let Ef (x) = k≥0 f (n)xn /n! be the exponential generating function of f. Theorem 3.1. Let #Y be the cardinality of a finite set Y . Given functions f, g : N → R, define a new function h : N → R by the rule X f (#S)g(#T ), h(#X) = (S,T )
GENERATING FUNCTIONS AND ROTA–BAXTER ALGEBRAS
5
where X is a finite set, and where (S, T ) ranges over all weak ordered partitions of X into two blocks, i.e., S ∩ T = ∅ and S ∪ T = X. Then (3.3)
Eh (x) = Ef (x)Eg (x). (λ)
We prove the following generalization of Theorem 3.1. For f : N → R, call Ef (x) :=
P
k≥0 f (n)1k ∈ d Xλ (R) be the λ-exponential generating function of f . When λ = 0, this recovers the exponential generating function.
Theorem 3.2. Given functions f, g : N → R, define a new function h : N → R by the rule X h(#X) = λ#(S∩T ) f (#S)g(#T ), (S,T )
where X is a finite set, and where (S, T ) ranges over all ordered subsets of X such that S ∪ T = X. Then (λ) (λ) Eh (x) = Ef (x)Eg(λ) (x) ∈ d Xλ (R).
(3.4)
Proof. We have (λ)
Ef (x)Eg(λ) (x)
X
=
f (m)1m
m≥0
=
X
X
n≥0
= by Eq. (3.2). Note that (λ) Ef (x)Eg(λ) (x)
m+n−k
= =
m
m k
∞ X u X m X
u=0 m=0 k=0 ∞ X u=0
f (m)g(n)
m,n
=
k
λ
f (m)g(n)1m 1n
m,n
X
g(n)1n
m X
i
λ
i=0
m+n−k m
m k
1m+n−k
k,m−k,m+n−2k
f (m)g(u − m + k) 1k
m+n−k
u k,m−k,u−m
X
u1
λ
(u1 ,u2 ,u3 )∈N3 ,u1 +u2 +u3 =u 3
Now the theorem follows since, for given (u1 , u2 , u3 ) ∈ N ,
. So setting u = m + n − k, we have
u u1 ,u2 ,u3
u u1 ,u2 ,u3
f (u1 + u2 )g(u1 + u3 ) 1u .
is the number of ways of partitioning
a size u set into subsets of size u1 , u2 and u3 . It is also the number of ways of taking subsets S of size u1 + u2 and T of size u1 + u3 of a X of size u1 + u2 + u3 such that S ∪ T = X and #(S ∩ T ) = u1 . 3.3. Stirling numbers. We next give an example of weight 1 exponential generating functions. Stirling numbers of the first kind, s(n, k), and the second kind, S(n, k), n, k ∈ N, are defined by (x)n =
n X
s(n, k)xk ,
xn =
k=0
n X
S(n, k)(x)k .
k=0
Stirling numbers of the second kind have the recursive formula S(n + 1, k + 1) = S(n, k) + (k + 1)S(n, k + 1), n, k ≥ 0 with S(0, 0) = 1, S(n, 0) = S(0, k) = 0 for n, k ≥ 1. It follows that S(n, 1) = S(n, n) = 1, n ≥ 1. The importance of these numbers in combinatorics is well-known [32]. For instance, S(n, k) is the number of partitions of n objects into k non-empty cells. They have the generating function (3.5)
exp[t(eu − 1)] =
∞ X n X
n=0 k=0
S(n, k)tk un /n!.
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Li Guo Theorem 3.3. We have the generating function e(1⊗1)u =
∞ X n X
n=0 k=0
S(n, k)k!1k un /n! ∈ d X1 (R)[[u]].
It has a much simpler form compared with Eq. (3.5). k!S(n, k) is the number of ways to put n different objects into m different non-empty cells, and is the number of surjective maps from {1, . . . , n} to {1, . . . , k}. 4. Free commutative Rota-Baxter algebras of weight 0 Let X be a set. Then the free commutative Rota-Baxter algebra of weight 0 X0 (X) := X0 (R[X]) is the vector space X0 (X) = ⊕n≥0 RX n
( identifying x1 ⊗ · · · ⊗ xn ↔ (x1 , · · · , xn ))
with the product defined by Eq. (2.2) with λ = 0. For example, (a0 ⊗ a1 ) (b0 ⊗ b1 ) = (a0 b0 ⊗ a1 ⊗ b1 ) + (a0 b0 ⊗ b1 ⊗ a1 ). In general (a0 ⊗ a1 ⊗ · · · ⊗ am ) (b0 ⊗ b1 ⊗ · · · ⊗ bn ) = a0 b0 ⊗
X
shuffles of (a1 ⊗ · · · ⊗ am ) and (b1 ⊗ · · · ⊗ bn ) .
We have the following examples of generating functions in the completion d X0 (X) of X0 (X). X X 1 = (1 ⊗ x)n = n!(1 ⊗ x⊗n ). 1−1⊗x n≥0
n≥0
X X (2n)! 1 = (1 ⊗ x ⊗ x)n = (1 ⊗ x⊗(2n) ). 1−1⊗x⊗x 2n 5. Free commutative Rota-Baxter algebras of weight 1 Let X1 (x) := X1 (R[x]) =
∞ M
⊗k
R[x]
=
k=1
∞ M k=1
R{xm m ≥ 1}k .
Recall that the product in X1 (x) and hence in d X1 (x) is given by the mixable shuffle product. For example, (a0 ⊗ a1 ⊗ a2 )(b0 ⊗ b1 ) = a0 b0 ⊗ a1 ⊗ a2 ⊗ b1 + a1 ⊗ b1 ⊗ a2 + a1 ⊗ a2 b1 + b1 ⊗ a1 ⊗ a2 + a1 b1 ⊗ a2 . Consider powers (1 ⊗ x)n ∈ X1 (x). We have (1 ⊗ x)2
= 2(1 ⊗ x ⊗ x) + 1 ⊗ x2 ,
(1 ⊗ x)3
= 6(1 ⊗ x ⊗ x ⊗ x) + 3(1 ⊗ x ⊗ x2 ) +3(1 ⊗ x2 ⊗ x) + (1 ⊗ x3 ).
4
(1 ⊗ x)
= 24(1 ⊗ x ⊗ x ⊗ x ⊗ x) + 12(1 ⊗ x ⊗ x ⊗ x2 ) + 12(1 ⊗ x ⊗ x2 ⊗ x) +12(1 ⊗ x2 ⊗ x ⊗ x) + 6(1 ⊗ x2 ⊗ x2 ) + 4(1 ⊗ x ⊗ x3 ) +4(1 ⊗ x3 ⊗ x) + (1 ⊗ x4 ).
Note that the terms on the right hand side for (1 ⊗ x)3 correspond to the = 1+ 1+1 = ordered partitions n 3 3 3 3 1 + 2 = 2 + 1 = 3 and the coefficients are the multi-nomial coefficients , , , . The 1,1,1
1,2
2,1
3
same is true for (1 ⊗ x)⊗4 . ⊗(2,3) 2 3 k In general, for I = (i1 , · · · , ik ) ∈ Nk , denote x⊗I = xi1 ⊗ · · · ⊗ xik . So x = x ⊗x . Let I = N>0 . Pk n n For I ∈ I, define the norm of I to be |I| = s=1 is , length ℓ(I) = k and = . I
i1 ,··· ,ik
GENERATING FUNCTIONS AND ROTA–BAXTER ALGEBRAS
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Theorem 5.1. We have the generating function ∞ ∞ X X X 1 n = (1 ⊗ x)n = (1 ⊗ x⊗I ) ∈ d X1 (x). I 1 − (1 ⊗ x) n=0 n=0 |I|=n |I| Here we have used the convention that, for I = ∅, take |I| = 0, = 1, (1 ⊗ x)I = 1 ⊗ x⊗I = 1. I
Proof. It follows from the following facts [19]. P n (1) (1 ⊗ x)n = |I|=n (1 ⊗ x⊗I ). I
(2) The sum is over the set of ordered partitions of n. X n (3) = k!S(n, k). |I|=n,ℓ(I)=k
I
References [1] M. Aguiar and S. Hsiao, Canonical characters on quasi-symmetric functions and bivariate Catalan numbers, Electron. J. Combin. 11(2) (2005), #R15, 34pp. [2] J. M. Borwein, D. J. Broadhurst, D. M. Bradley, and P. Lisonˇ ek, Special values of multiple polylogarithms, Trans. Amer. Math. Soc., 353, (2001), no. 3, 907–941. arXiv:math.CA/9910045 [3] D. M. Bradley, Partition identities for the multiple zeta function, in “Zeta Functions, Topology, and Quantum Physics” T. Aoki et. al. (eds.), Springer, 2005, 19–29. [4] D. M. Bradley, Multiple q-zeta values, J. Algebra, 283, (2005), no. 2, 752–798 arXiv:math.QA/0402093 [5] D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, 393, (1997), no. 3-4, 403–412. [6] P. Cartier, On the structure of free Baxter algebras, Adv. in Math., 9, (1972), 253–265. [7] P. Cartier, Fonctions polylogarithmes, nombres polyztas et groupes pro-unipotents, Astrisque, 282, (2002), 137–173, (Sm. Bourbaki no. 885). [8] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 210, (2000), no. 1, 249–273. arXiv:hepth/9912092 [9] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 216, (2001), no. 1, 215–241. arXiv:hep-th/0003188 [10] K. Ebrahimi-Fard and L. Guo, Mixable Shuffles, Quasi-shuffles and Hopf Algebras, J. Alg. Combinatorics, 24 (2006), 83–101, arXiv:math.RA/0506418. [11] K. Ebrahimi-Fard and L. Guo, Multiple zeta values and Rota-Baxter algebras, preprint, arXiv:math.NT/0601558. [12] K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable Renormalization II: the General case, Annales Henri Poincar´ e, 6, (2005), 369–395. arXiv:hep-th/0403118 [13] K. Ebrahimi-Fard, L. Guo and D. Kreimer, Spitzer’s Identity and the Algebraic Birkhoff Decomposition in pQFT, J. Phys. A: Math. Gen., 37, (2004) 11037–11052. arXiv: hep-th/0407082 [14] R. Ehrenborg, On postes and Hopf algebras, Adv. Math. 119 (1996), 1-25. [15] F. Fares, Quelques constructions d’alg` ebres et de coalg` ebres, Universit´ e du Qu´ ebec a ` Mont´ eal (1999). [16] A. G. Goncharov, Periods and mixed motives, arXiv:math.AG/0202154. [17] L. Guo, Baxter algebra and differential algebra, in: “Differential Algebra and Related Topics”, World Scientific Publishing Company, (2002), 281–305. [18] L. Guo, Baxter algebras and the umbral calculus, Adv. in Appl. Math., 27, (2001), no. 2-3, 405–426. [19] L. Guo, Baxter algebras, Stirling numbers and partitions, J. Algebra Appl., 4, (2005), no. 2, 153–164. arXiv:math.AC/0402348 [20] L. Guo and W. Keigher, Free Baxter algebras and shuffle products, Adv. in Math., 150, (2000), 117–149. [21] L. Guo and W. Keigher, On Baxter algebras: completions and the internal construction, Adv. in Math., 151, (2000), 101–127. [22] L. Guo and W. Y. Sit, Enumeration of Rota-Baxter words, in Proceedings ISSAC 2006, Genoa, Italy, ACM Press, arXiv: math.RA/0602449. [23] L. Guo and B. Zhang, Renormalization of multiple zeta values, arXiv:math.NT/0606076. [24] M. Hazewinkel, Generalized overlapping shuffle algebras J. Math. Sci. (New York), 106 (2001), 3168-3186. [25] M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra, 194, no. 2, (1997), 477–495. [26] M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin., 11, no. 1, (2000), 49–68. [27] W. Keigher, On the ring of Hurwitz series, Comm. Algebra, 25 (1997), 1845-1859. [28] J.-L. Loday, On the algebra of quais-shuffles, to appear in Manuscripta Mathematica, arXiv: math.QA/0506498.
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Li Guo [29] D. Manchon and S. Paycha, Renormalized Chen integrals for symbols on Rn and renormlized polyzeta functions, arXiv:math.NT/0604562. [30] G.-C. Rota, Baxter algebras and combinatorial identities I & II, Bull. Amer. Math. Soc., 75, (1969), 325–329, 330–334. [31] G.-C. Rota and D. A. Smith, Fluctuation theory and Baxter algebras, Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilit, INDAM, Rome, 1971), pp. 179–201. Academic Press, London, (1972). [32] R. Stanley, Enumerative Combinatorics, Vol 1-2, Cambridge University Press, 1999. [33] H. Wilf, Generatingfunctionology, available on line at his website at www.math.upenn.edu Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102 E-mail address:
[email protected] URL: http://newark.rutgers.edu/~liguo
